↑

Properties

Label 8-99e4-1.1-c3e4-0-1
Degree $8$
Conductor $96059601$
Sign $1$
Analytic cond. $1164.13$
Root an. cond. $2.41685$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s + 16·4-s + 54·5-s + 27·9-s + 128·12-s + 432·15-s + 64·16-s + 864·20-s + 1.49e3·25-s + 136·27-s − 340·31-s + 432·36-s − 868·37-s + 1.45e3·45-s + 108·47-s + 512·48-s − 686·49-s + 2.16e3·59-s + 6.91e3·60-s − 1.02e3·64-s + 416·67-s + 1.19e4·75-s + 3.45e3·80-s + 1.08e3·81-s − 2.72e3·93-s − 34·97-s + 2.39e4·100-s + ⋯
L(s)  = 1  + 1.53·3-s + 2·4-s + 4.82·5-s + 9-s + 3.07·12-s + 7.43·15-s + 16-s + 9.65·20-s + 11.9·25-s + 0.969·27-s − 1.96·31-s + 2·36-s − 3.85·37-s + 4.82·45-s + 0.335·47-s + 1.53·48-s − 2·49-s + 4.76·59-s + 14.8·60-s − 2·64-s + 0.758·67-s + 18.4·75-s + 4.82·80-s + 1.49·81-s − 3.03·93-s − 0.0355·97-s + 23.9·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(96059601\)    =    \(3^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1164.13\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 96059601,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(25.47639716\)
\(L(\frac12)\) \(\approx\) \(25.47639716\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2^2$ \( 1 - 8 T + 37 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
good2$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
5$C_2$$\times$$C_2^2$ \( ( 1 - 18 T + p^{3} T^{2} )^{2}( 1 - 18 T + 199 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} ) \)
7$C_2^2$ \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
19$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
23$C_2^2$$\times$$C_2^2$ \( ( 1 - 108 T - 503 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} )( 1 + 108 T - 503 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} ) \)
29$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
31$C_2$$\times$$C_2^2$ \( ( 1 + 340 T + p^{3} T^{2} )^{2}( 1 - 340 T + 85809 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} ) \)
37$C_2^2$ \( ( 1 + 434 T + 137703 T^{2} + 434 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 - 36 T + p^{3} T^{2} )^{2}( 1 - 36 T - 102527 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} ) \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 738 T + 395767 T^{2} - 738 p^{3} T^{3} + p^{6} T^{4} )( 1 + 738 T + 395767 T^{2} + 738 p^{3} T^{3} + p^{6} T^{4} ) \)
59$C_2$$\times$$C_2^2$ \( ( 1 - 720 T + p^{3} T^{2} )^{2}( 1 - 720 T + 313021 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} ) \)
61$C_2^2$ \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 - 416 T + p^{3} T^{2} )^{2}( 1 + 416 T - 127707 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4} ) \)
71$C_2^2$$\times$$C_2^2$ \( ( 1 - 612 T + 16633 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} )( 1 + 612 T + 16633 T^{2} + 612 p^{3} T^{3} + p^{6} T^{4} ) \)
73$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 1674 T + p^{3} T^{2} )^{2}( 1 + 1674 T + p^{3} T^{2} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + 34 T + p^{3} T^{2} )^{2}( 1 - 34 T - 911517 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.01986700992271642660004095014, −9.391543088172518802431851333448, −9.086875298292518709151657611125, −9.009486007175137140161306576501, −8.962963624595547640413915780565, −8.389186593565339933640920635511, −8.160060721975779114060267801033, −7.51854908375264933207533244159, −7.09399896816403730068622128522, −6.95848809238039879183337017649, −6.58219512779800012654329297619, −6.47650992852205432959200102295, −6.25651809488044491290577328758, −5.68171691590687687259001999453, −5.37850711380333243740967856330, −5.16516885953056132747176785510, −5.16043172059926993059378531720, −3.97570366464804434049573900292, −3.48390523821825680201061269882, −2.92080842682733292288340224907, −2.58390832135917392322128372153, −2.21930953843618595899237606192, −2.04638338869995995682604850446, −1.66286287760416819673824352355, −1.48181067170182188379870680000, 1.48181067170182188379870680000, 1.66286287760416819673824352355, 2.04638338869995995682604850446, 2.21930953843618595899237606192, 2.58390832135917392322128372153, 2.92080842682733292288340224907, 3.48390523821825680201061269882, 3.97570366464804434049573900292, 5.16043172059926993059378531720, 5.16516885953056132747176785510, 5.37850711380333243740967856330, 5.68171691590687687259001999453, 6.25651809488044491290577328758, 6.47650992852205432959200102295, 6.58219512779800012654329297619, 6.95848809238039879183337017649, 7.09399896816403730068622128522, 7.51854908375264933207533244159, 8.160060721975779114060267801033, 8.389186593565339933640920635511, 8.962963624595547640413915780565, 9.009486007175137140161306576501, 9.086875298292518709151657611125, 9.391543088172518802431851333448, 10.01986700992271642660004095014

Graph of the $Z$-function along the critical line